Math worksheet focusing on solving multi-step inequalities with fractions, including problems with variables in numerators and denominators, and instructions to graph the solutions.

Let's solve each of the multi-step inequalities with fractions step by step and then explain how to graph the solution.

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① $\frac{3}{5}(2d - 5) \leq 4\left(7 - \frac{1}{5}d\right)$

#### Step 1: Expand both sides
Left:
$$
\frac{3}{5}(2d - 5) = \frac{6d}{5} - 3
$$
Right:
$$
4\left(7 - \frac{1}{5}d\right) = 28 - \frac{4}{5}d
$$

So the inequality becomes:
$$
\frac{6d}{5} - 3 \leq 28 - \frac{4d}{5}
$$

#### Step 2: Eliminate fractions by multiplying both sides by 5 (LCM)
$$
5\left(\frac{6d}{5} - 3\right) \leq 5\left(28 - \frac{4d}{5}\right)
$$
$$
6d - 15 \leq 140 - 4d
$$

#### Step 3: Solve for $d$
Add $4d$ to both sides:
$$
10d - 15 \leq 140
$$
Add 15:
$$
10d \leq 155
$$
Divide by 10:
$$
d \leq 15.5
$$

✔ Solution: $d \leq 15.5$

#### Graph:
- Draw a number line.
- Place a closed circle at $15.5$ (since it’s “≤”).
- Shade to the left.

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② $\frac{6d}{5} - 3 + 3 \leq 28 - \frac{4d}{5} + 3$

Simplify both sides:

Left: $\frac{6d}{5} - 3 + 3 = \frac{6d}{5}$
Right: $28 + 3 - \frac{4d}{5} = 31 - \frac{4d}{5}$

So:
$$
\frac{6d}{5} \leq 31 - \frac{4d}{5}
$$

Multiply both sides by 5:
$$
6d \leq 155 - 4d
$$

Add $4d$:
$$
10d \leq 155
$$

Divide by 10:
$$
d \leq 15.5
$$

✔ Same as #1: $d \leq 15.5$

Graph same as above.

---

③ $\frac{2x - 3}{4} + 9 \geq 3 + \frac{4x}{3}$

#### Step 1: Eliminate denominators — LCM of 4 and 3 is 12

Multiply every term by 12:
$$
12 \cdot \left(\frac{2x - 3}{4}\right) + 12 \cdot 9 \geq 12 \cdot 3 + 12 \cdot \left(\frac{4x}{3}\right)
$$
$$
3(2x - 3) + 108 \geq 36 + 16x
$$

Expand:
$$
6x - 9 + 108 \geq 36 + 16x
$$
$$
6x + 99 \geq 36 + 16x
$$

Subtract $6x$:
$$
99 \geq 36 + 10x
$$

Subtract 36:
$$
63 \geq 10x
$$

Divide by 10:
$$
x \leq 6.3
$$

✔ Solution: $x \leq 6.3$

Graph: Closed circle at $6.3$, shade left.

---

④ $\frac{3x + 1}{x - 1} - 2 \geq 0$

#### Step 1: Combine terms
$$
\frac{3x + 1}{x - 1} - 2 \geq 0
\Rightarrow \frac{3x + 1 - 2(x - 1)}{x - 1} \geq 0
$$

Simplify numerator:
$$
3x + 1 - 2x + 2 = x + 3
$$

So:
$$
\frac{x + 3}{x - 1} \geq 0
$$

#### Step 2: Critical points: $x = -3$, $x = 1$ (denominator zero → undefined)

Use sign chart:

| Interval | Sign of $(x+3)$ | Sign of $(x-1)$ | Overall |
|------------------|------------------|------------------|---------|
| $x < -3$ | – | – | + |
| $-3 < x < 1$ | + | – | – |
| $x > 1$ | + | + | + |

We want where expression ≥ 0 → positive or zero.

So:
- $x \leq -3$ (includes $x = -3$, since numerator is 0)
- $x > 1$ (but not $x = 1$, undefined)

✔ Solution: $x \leq -3$ or $x > 1$

Graph:
- Closed circle at $-3$, open at $1$
- Shade from $-\infty$ to $-3$, and from $1$ to $\infty$

---

⑤ $2\left(\frac{2x + 2}{4} - \frac{3x - 3}{6}\right) < 2x - 4$

#### Step 1: Simplify inside parentheses

First, simplify each fraction:
$$
\frac{2x + 2}{4} = \frac{2(x + 1)}{4} = \frac{x + 1}{2}
$$
$$
\frac{3x - 3}{6} = \frac{3(x - 1)}{6} = \frac{x - 1}{2}
$$

Now:
$$
2\left(\frac{x + 1}{2} - \frac{x - 1}{2}\right) < 2x - 4
$$

Combine:
$$
2\left(\frac{(x + 1) - (x - 1)}{2}\right) = 2\left(\frac{2}{2}\right) = 2(1) = 2
$$

So:
$$
2 < 2x - 4
$$

Add 4:
$$
6 < 2x
$$

Divide by 2:
$$
3 < x \quad \text{or} \quad x > 3
$$

✔ Solution: $x > 3$

Graph: Open circle at 3, shade to the right.

---

⑥ $-9 > -\frac{1}{3}x + 6$

#### Step 1: Subtract 6
$$
-9 - 6 > -\frac{1}{3}x
\Rightarrow -15 > -\frac{1}{3}x
$$

#### Step 2: Multiply both sides by $-3$ → flip inequality
$$
(-3)(-15) < x \Rightarrow 45 < x \Rightarrow x > 45
$$

✔ Solution: $x > 45$

Graph: Open circle at 45, shade right.

---

⑦ $\frac{3x + 4}{3} - \frac{15x}{3} < 6$

Note: Both terms over 3 → combine numerators:
$$
\frac{3x + 4 - 15x}{3} < 6 \Rightarrow \frac{-12x + 4}{3} < 6
$$

Multiply both sides by 3:
$$
-12x + 4 < 18
$$

Subtract 4:
$$
-12x < 14
$$

Divide by $-12$ → flip inequality
$$
x > -\frac{14}{12} = -\frac{7}{6}
$$

✔ Solution: $x > -\frac{7}{6} \approx -1.166...$

Graph: Open circle at $-\frac{7}{6}$, shade right.

---

⑧ $\frac{3}{5}x - 3 \geq \frac{3}{10}x - 9$

#### Step 1: Eliminate fractions — LCM of 5 and 10 is 10

Multiply all terms by 10:
$$
10\left(\frac{3}{5}x\right) - 10(3) \geq 10\left(\frac{3}{10}x\right) - 10(9)
\Rightarrow 6x - 30 \geq 3x - 90
$$

Subtract $3x$:
$$
3x - 30 \geq -90
$$

Add 30:
$$
3x \geq -60
$$

Divide by 3:
$$
x \geq -20
$$

✔ Solution: $x \geq -20$

Graph: Closed circle at $-20$, shade right.

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## ✔ Final Answers Summary:

| Problem | Solution |
|--------|----------|
| ① | $d \leq 15.5$ |
| ② | $d \leq 15.5$ |
| ③ | $x \leq 6.3$ |
| ④ | $x \leq -3$ or $x > 1$ |
| ⑤ | $x > 3$ |
| ⑥ | $x > 45$ |
| ⑦ | $x > -\frac{7}{6}$ |
| ⑧ | $x \geq -20$ |

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## 🔍 How to Graph Each:

For each inequality:
- Use closed circle for $\leq$, $\geq$
- Use open circle for $<$, $>$
- Shade in the direction that satisfies the inequality
- For rational expressions (like #4), exclude values that make denominator zero

Let me know if you'd like a visual representation of any graph!